Problem 70 - University Physics with Modern Physics

Question

A canister of 1.20 mol of nitrogen gas \((28.0 \mathrm{g} / \mathrm{mol})\) at \(25.0^{\circ} \mathrm{C}\) is left on Jupiter's satellite after completion of a future space mission. Europa has no appreciable atmosphere, and the acceleration due to gravity at its surface is 1.30 \(\mathrm{m} / \mathrm{s}^{2}\) . After some time, the canister springs a small leak, allowing molecules to escape through a small hole. What is the maximum height (in \(\mathrm{km}\) ) above Europa's surface that a \(\mathrm{N}_{2}\) molecule having speed equal to the rms speed will reach if it is shot straight up out of the hole in the canister? Ignore the variation in \(g\) with altitude,

Step-by-Step Solution

Answer

The maximum height is approximately 102.8 km.

Step 1: Calculate the rms speed of N2 molecules

First, calculate the root mean square (rms) speed using the formula for rms speed: \[ v_{rms} = \sqrt{\frac{3k_B T}{m}} \] where \(k_B\) is the Boltzmann constant \((1.38 \times 10^{-23} \text{ J/K})\), \(T\) is the temperature in Kelvin (convert 25.0°C to 298 K), and \(m\) is the mass of one molecule of \(\text{N}_2\) (molar mass of nitrogen is 28.0 g/mol or 28.0 kg/kmol, so the mass of one molecule \(= \frac{28.0 \times 10^{-3}}{6.022 \times 10^{23}}\)).\Calculated rms speed: \[ v_{rms} \approx \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 298}{\frac{28.0 \times 10^{-3}}{6.022 \times 10^{23}}}} \approx 517 \text{ m/s} \]

Step 2: Use energy conservation to determine maximum height

Apply the conservation of mechanical energy principle. The total mechanical energy at the surface, which is kinetic energy due to rms speed, will equal the potential energy at maximum height.\\[ \frac{1}{2} m v_{rms}^2 = m g h_{max} \]\Solving for \(h_{max}\), we get: \[ h_{max} = \frac{v_{rms}^2}{2g} \] where \(g = 1.30 \text{ m/s}^2\) is Europa's gravity.\[ h_{max} = \frac{517^2}{2 \times 1.30} = \frac{267289}{2.6} \text{ meters} \approx 102804 \text{ meters} = 102.804 \text{ km} \]

Key Concepts

Kinetic Energy

Kinetic energy is an essential concept in the understanding of motion. It refers to the energy that an object possesses due to its motion. - Formula: The kinetic energy (\[ KE \] ) of an object with mass (\[ m \] ) and moving with velocity (\[ v \] ) is given by the equation:\[ KE = \frac{1}{2} mv^2 \]- Direction doesn't matter: Whether an object is moving up, down, or sideways, kinetic energy is always calculated the same way.In the context of the nitrogen molecule in the exercise, we use the equation to determine how high the molecule can travel. At the start, the molecule’s kinetic energy due to its speed needs to be enough to overcome the gravitational pull of Europa, allowing the molecule to rise to a maximum height. By equating this kinetic energy to the gravitational potential energy at the highest point, we can estimate how far the molecule can jump if it escapes in the canister's direction.

Root Mean Square Speed

Root mean square (RMS) speed is an important concept in the kinetic theory of gases. It is a measure used to describe the speed of particles in a gas, which is related to the average kinetic energy of the gas's molecules.- Formula: The RMS speed is given by:\[ v_{rms} = \sqrt{\frac{3k_B T}{m}} \]where: - \( v_{rms} \) is the root mean square speed - \( k_B \) is the Boltzmann constant - \( T \) is the absolute temperature in Kelvin - \( m \) is the mass of one molecule- RMS in the exercise: For the nitrogen molecules escaping from the canister, the RMS speed is crucial because it provides an average speed that helps in determining how high a molecule can go when escaping Europa's gravitational field. This speed provides a way to link thermal energy in the gas to the mechanical motion of individual nitrogen molecules.

Conservation of Energy

The principle of conservation of energy is a fundamental concept in physics. It states that energy cannot be created or destroyed in an isolated system, only transformed from one form to another.- In mechanics, this principle often involves the transformation between kinetic energy and potential energy.In the problem, conservation of energy is key to finding the height that a nitrogen molecule can reach. Initially, the nitrogen molecule's energy is all kinetic due to its movement which derives from thermal motion. As it rises, kinetic energy is converted into gravitational potential energy until the molecule reaches its max height, where the kinetic energy is zero. This principle allows us to set up an equation where the initial kinetic energy equals the final potential energy:\[ \frac{1}{2} m v_{rms}^2 = mgh_{max} \]This relationship allows us to solve for the maximum height.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It depends on the object's mass, the gravitational field strength, and the height above a reference point.- Formula: The potential energy (\[ U \] ) due to gravity is:\[ U = mgh \]where: - \( m \) is the mass - \( g \) is the gravitational acceleration - \( h \) is the height above the surfaceIn the given problem, as the nitrogen molecule moves upwards from the surface of Europa, its kinetic energy is converted into gravitational potential energy. The farther it rises, the more gravitational potential energy it gains and the less kinetic energy it has, until all the kinetic energy is converted, determining the maximum height. Understanding this concept helps us see how the molecule's speed and the gravitational pull of Europa balance each other to define how high it can go.